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A Lot of Examples of Generalized Weak Bi-Frobenius Algebras

Received: 23 February 2019     Accepted: 4 April 2019     Published: 26 April 2019
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Abstract

In this paper, by considering the tensor product of a bi-Frobenius algebra and a weak Hopf algebra, a lot of examples of the generalized weak bi-Frobenius algebras are given, such as the 16-dimensional, 24-dimensional and 40-dimensional GWBF algebras. They provide a common generalization of weak Hopf algebras introduced by Böhm, Nill, Szlachányi, and of bi-Frobenius algebras introduced by Doi and Takeuchi.

Published in International Journal of Discrete Mathematics (Volume 4, Issue 1)
DOI 10.11648/j.dmath.20190401.16
Page(s) 38-44
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Examples, Bi-Frobenius Algebras, Generalized Weak Bi-Frobenius Algebras

References
[1] Y. Doi. Substructures of bi-Frobenius algebras. J. Algebra 256 (2), 568-582, 2002.
[2] Y. Doi, M. Takeuchi. Bi-Frobenius algebras. New trends in Hopf algebra theory (La Falda, 1999), 67-97, Contemp. Math., 267, Amer. Math. Soc., Providence, RI, 2000.
[3] Q. G. Chen, S. H. Wang. Radford’s formula for generalized weak biFrobenius algebras. Rocky Mountain J. Math. 44 (2), 419-433, 2014.
[4] J. Bichon. The group of bi-Galois objects over the coordinate algebra of the Frobenius-Lusztig kernel of SL (2). Glasg. Math. J. 58 (3), 727-738, 2016.
[5] Y. Y. Chen, L. Y. Zhang. The structure and construction of bi-Frobenius Hom-algebras. Comm. Algebra 45 (5), 2142-2162, 2017.
[6] Z. H. Wang, L. B. Li. Double Frobenius algebras. Front. Math. China 13 (2), 399-415, 2018.
[7] Y. H. Wang, X. W. Chen. Construct non-graded bi-Frobenius algebras via quivers. Sci. China Ser. A 50 (3), 450-456, 2007.
[8] G. Böhm, F. Nill, K. Szlachányi. Weak Hopf Algebras - I. Integral Theory and C* -Structure. J. Algebra 221, 385-438, 1999.
[9] D. Nikshych. Semisimple weak Hopf algebras. J. Algebra 275 (2), 639-667, 2004.
[10] D. Bulacu. A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category. J. Algebra 332, 244–284, 2011.
[11] H. X. Zhu. The quantum double of a factorizable weak Hopf algebra. Comm. Algebra 45 (9), 4067-4083, 2017.
[12] Y. H. Wang. Braided bi-Frobenius algebras. (Chinese) Chinese Ann. Math. Ser. A 28 (2), 203-214, 2007.
[13] Y. H. Wang, P. Zhang. Construct bi-Frobenius algebras via quivers. Tsukuba J. Math. 28 (1), 215-221, 2004.
[14] N. Yamatan. S4-formula and S2-formula for quasi-triangular bi-Frobenius algebras. Tsukuba J. Math. 26 (2), 339-349, 2002.
[15] C. Kassel. Quantum Groups, Springer-Verlag, New York, 1995.
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    Yan Sun, Xiaohui Zhang. (2019). A Lot of Examples of Generalized Weak Bi-Frobenius Algebras. International Journal of Discrete Mathematics, 4(1), 38-44. https://doi.org/10.11648/j.dmath.20190401.16

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    ACS Style

    Yan Sun; Xiaohui Zhang. A Lot of Examples of Generalized Weak Bi-Frobenius Algebras. Int. J. Discrete Math. 2019, 4(1), 38-44. doi: 10.11648/j.dmath.20190401.16

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    AMA Style

    Yan Sun, Xiaohui Zhang. A Lot of Examples of Generalized Weak Bi-Frobenius Algebras. Int J Discrete Math. 2019;4(1):38-44. doi: 10.11648/j.dmath.20190401.16

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  • @article{10.11648/j.dmath.20190401.16,
      author = {Yan Sun and Xiaohui Zhang},
      title = {A Lot of Examples of Generalized Weak Bi-Frobenius Algebras},
      journal = {International Journal of Discrete Mathematics},
      volume = {4},
      number = {1},
      pages = {38-44},
      doi = {10.11648/j.dmath.20190401.16},
      url = {https://doi.org/10.11648/j.dmath.20190401.16},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20190401.16},
      abstract = {In this paper, by considering the tensor product of a bi-Frobenius algebra and a weak Hopf algebra, a lot of examples of the generalized weak bi-Frobenius algebras are given, such as the 16-dimensional, 24-dimensional and 40-dimensional GWBF algebras. They provide a common generalization of weak Hopf algebras introduced by Böhm, Nill, Szlachányi, and of bi-Frobenius algebras introduced by Doi and Takeuchi.},
     year = {2019}
    }
    

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    AB  - In this paper, by considering the tensor product of a bi-Frobenius algebra and a weak Hopf algebra, a lot of examples of the generalized weak bi-Frobenius algebras are given, such as the 16-dimensional, 24-dimensional and 40-dimensional GWBF algebras. They provide a common generalization of weak Hopf algebras introduced by Böhm, Nill, Szlachányi, and of bi-Frobenius algebras introduced by Doi and Takeuchi.
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Author Information
  • School of Mathematical Sciences, Qufu Normal University, Qufu, The People's Republic of China

  • School of Mathematical Sciences, Qufu Normal University, Qufu, The People's Republic of China

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